Question

Two generators have cost functions with incremental-cost characteristics: 
\[ \frac{dF_1}{dP_1} = 40 + 0.2 P_1, \frac{dF_2}{dP_2} = 32 + 0.4 P_2 \] They must supply a total load of 260 MW. Find the optimal generation (economic dispatch) ignoring losses. 
 

• A. \(P_1 = P_2 = 130\)
• B. \(P_1 = 160,\; P_2 = 100\)
• C. \(P_1 = 140,\; P_2 = 120\)
• D. \(P_1 = 120,\; P_2 = 140\)

Hint:

In economic dispatch without losses, equate incremental costs and apply power balance to get optimal generation.

Answer 1

The Correct Option is B

Step 1: Economic dispatch condition.
For optimal operation (no losses), incremental costs must be equal:
\[ 40 + 0.2P_1 = 32 + 0.4P_2 \]

Step 2: Use total power constraint.
\[ P_1 + P_2 = 260 \]

Step 3: Solve the system.
Subtract constants: \[ 40 - 32 = 0.4P_2 - 0.2P_1 \] \[ 8 = 0.4P_2 - 0.2P_1 \] Substitute \(P_2 = 260 - P_1\):
\[ 8 = 0.4(260 - P_1) - 0.2P_1 \] \[ 8 = 104 - 0.4P_1 - 0.2P_1 \] \[ 0.6P_1 = 96 \] \[ P_1 = 160, P_2 = 100 \]

Step 4: Conclusion.
Only option (B) satisfies both economic dispatch and total load constraints.

Final Answer: \(\boxed{P_1 = 160,\; P_2 = 100}\)